Quantum Gravity Seminar

Week 15, Track 2

John Baez

February 12, 2001

Oz hesitated... should he keep secretly attending Track 2? On the one
hand, he was having a lot of trouble following the material. On the
other hand, how would he ever learn this stuff if he didn't try? He
stood pondering the matter until he heard the Wiz and his Acolytes
nearing the classroom. At the last minute, more out of panic than
anything else, he ducked under a desk.

The Wiz started in: "So far we've been studying 2d electromagnetism in a
very rigid way. First we fixed a topology for spacetime: the cylinder.
Then we fixed a metric on spacetime. Then we picked a way to foliate
spacetime with spacelike slices. We proceeded from there... and found
that the theory was isomorphic to the classical mechanics of a free
point particle.

But now let's loosen things up! Let's figure out how the theory
works with an arbitrary 2-dimensional manifold as spacetime, with
an arbitrary metric and an arbitrary slicing.

To do this, we'll make use of a wonderful fact: when the dimension
of spacetime is 2, the vacuum Maxwell theory is almost background-free!"

"Heh," said the Wiz, turning to Toby and smiling.
"Indeed, we often think of background dependence as a yes-or-no
business. We say a field theory is "background-dependent" if
the Lagrangian involves some fields that are held fixed -- not varied
when we work out the equations of motion. These fields are called
"background fields". The classic example is the metric, which
we fix from the very start in ordinary special-relativistic quantum
field theory. We call this a "background metric". So far,
we've been using a background metric all over the place in our study of
2d electromagnetism.

When there are background fields around, they usually break diffeomorphism
invariance: not all diffeomorphims of spacetime act as symmetries of our
theory, but only those that preserve the background fields. For example,
in ordinary quantum field theory, the diffeomorphism group gets broken down
to the subgroup that preserves the Minkowski metric: the Poincare group.

Now, in our study of the 2d vacuum Maxwell theory, we started out by fixing
a metric on the cylinder:

g = dt2 - dx2

But in fact, we don't really need the metric in all its full glory!
We only need the volume form:

vol = dt ^ dx.

This counts as "less" background structure in the following precise
sense: the group of diffeomorphisms that preserve the volume form
contains the group of diffeomorphisms that preserve the metric.
So you see, we can really talk about "more" or
"less" background structure!

In fact, the vacuum Maxwell equations in 2d rely on so little
background structure that we can easily modify them and get some
truly background-free theories. This also works starting
from 2d Yang-Mills theory. That's the direction we're heading....

But I'm getting ahead of myself. Let's see why the vacuum Maxwell
equations in 2d only depend on the volume form. In any dimension,
we can write them as

dA = F
d*F = 0

The only role of the background metric is to define *F. But in 2
dimensions we have

F = e vol

for some function e called the "electric field", and then

*F = -e.

This lets us rewrite the vacuum Maxwell equations in a way that
uses only the volume form as a background structure:

dA = e vol
de = 0

We can also write down the Lagrangian in a way that only involves
the volume form! Normally we write it as

L = F ^ *F

but in 2d we can write it as

L = e2 vol.

Now, in 2 dimensions the volume form should really be called the "area
form", since it's what we use to measure areas. It follows from what
we've done that in 2d, vacuum electromagnetism has all *area-preserving*
diffeomorphisms as symmetries. These form a really big group. To get
a feeling for it, imagine an incompressible fluid in 2 dimensions. As
it flows, any particle that starts at any point x will move to some new
point x(t) at time t. This map

x |-> x(t)

is an area-preserving diffeomorphism. If you visualize a 2d fluid
swirling around, you'll see how wiggly and flexible such diffeomorphisms
can be!"

"In fact," said Miguel, "just as vector fields form the
Lie algebra of the diffeomorphism group, divergence-free
vector fields form the Lie algebra of this group of area-preserving
diffeomorphisms!"

"Right -- at least if one takes care to deal with these
infinite-dimensional groups and Lie algebras correctly," said the Wiz.

"And in V. I. Arnold's book, he shows that the equations for an
incompressible fluid with zero viscosity say that --"

Let's describe time evolution in 2d electromagnetism when spacetime is a
cylinder with an arbitrary Lorentzian metric. To be specific, let's do
the version of electromagnetism with gauge group R -- the U(1)
version will be similar.

We can slice spacetime any way we want... so we've got a bumpy
cylinder with two wiggly slices S and S':

The region of spacetime between S and S' is a cobordism M: S -> S', but
with a volume form on it.

We've seen that classically, a state of electromagnetism on the slice S
is described by a pair of numbers (a,e). The number a is the integral of
the A field around S, and e is the electric field.
Similarly, the state on the slice S' is some pair (a',e'). Time evolution
is described by the map

(a,e) |-> (a',e').

But what is this map like?

Well, Maxwell's equation in 2d says the electric field is constant, so

e' = e.

To figure out a', we use Stokes' theorem:

\intM dA = \intdM A.

This says that

\intM e vol = a' - a.

The left side is just e times the area of M, so

a' = a + Area(M) e.

In short, time evolution only depends on the area between our slices.
That's just what you'd expect in a 2d theory where the only background
field is the volume form... it really couldn't work any other way.

In fact, in 2d electromagnetism, area plays a role very much like
"time" in ordinary mechanics. If we use "t" to
stand for the area between our slices, our time evolution map is

(a,e) |-> (a + te, e).

This is isomorphic to time evolution for a point particle with unit
mass:

(q,p) |-> (q + tp, p).

Sound familiar? It should! We saw a special case of this in Week
13. And just as we did back then, we can use this isomorphism to quickly
figure out the quantum version of our theory. The Hilbert
space is L2(R), the Hamiltonian is copied after the
Hamiltonian for a particle of unit mass:

(H psi)(x) = -(1/2) psi''(x)

and to evolve a state in time from one slice to another, we just hit it
with the operator exp(-itH), where t is the area between the slices.

This whole business also works for the U(1) version of electromagnetism.
The only difference is that the holonomy a takes values in U(1), so our
Hilbert space is L2(U(1)). The formula for the Hamiltonian
looks just the same... no big deal.

Okay, great -- we've quantized 2d electromagnetism on a cylinder with
arbitrary metric. What's left to do?"

"More general manifolds," said Richard.

"Right! We'd really like to describe time evolution for any cobordism
between 1-dimensional manifolds, like this:

If we do this right, we'll get something almost like a 2d TQFT!
We'll get time evolution operators

Z(M): Z(S) -> Z(S')

satisfying a lot of the rules that hold in a TQFT. The main difference
is that now our cobordism M will have a specified area, and our time
evolution operator will depend on this area, not just the topology of M.

Before we can do this, though, there's a little problem we must
confront: the problem of "topology change". It's not easy to
put nice Lorentzian metrics on cobordisms that go between spaces with
different topologies. For example, there's no way to put a Lorentzian
metric on this manifold M:

for which S and S' are spacelike. Now, there are different ways to
tackle this problem. One is to allow degenerate Lorentzian metrics.
That's probably the best approach. But another is to pull a dirty
trick: switch to spacetimes equipped with a Riemannian metric instead
of a Lorentzian one! Let me explain this trick, because it's quite
popular.

First consider the cylinder R x S1.
Formally, we can turn the Lorentzian metric

-dt2 + dx2

into the Riemannian metric

dt2 + dx2

by means of this substitution:

t -> -it

This is called a Wick rotation. If we pull this trick, Schroedinger's
equation turns into the heat equation --"

Oz was very puzzled, so he interrupted, again mimicking Toby's voice:
"What do you mean by formally here?"

"It means: don't ask me what I mean.
Physicists say this when they
don't really know what they're doing."

"What kind of answer is that???" said Miguel. "Don't
tell me you're one of those guys who switch to imaginary time as soon as
the going gets tough, and then never come back to the real world."

"Yeah!" said Toby. "I've never been convinced that this
is a legitimate way to deal with topology change."

"Yeah!" said Miguel. "Sure, you can relate
Schroedinger's equation and the heat equation in situations when there's
a fixed time coordinate t around: you can take your formulas,
analytically continue them to imaginary values of t, do whatever you
want with them, and analytically continue back -- that's fine. But a
fixed time coordinate is exactly what you don't have in quantum
gravity. Especially in situations with topology change."

"Yeah!" said Toby. "This sucks! I want my money
back!" And without exchanging another word, Toby and Miguel stood
up and advanced towards the Wiz, brandishing their staffs menacingly.
"DOWN WITH WICK ROTATION!" they chanted... and the rest of
the class began to join in.

"Wait a minute!" cried the Wizard, backing into a corner.

"DOWN WITH WICK ROTATION!"

"Be patient!" he yelped, as the class began to wave pitchforks in the
air.

"DOWN WITH WICK ROTATION!"

"Listen: I AGREE WITH YOU! I'm not one of those evil physicists who
permanently flees the Lorentzian world for the imaginary paradise
of Riemannian spacetime. I just wanted to show you how it works!"

The class lowered their weapons, grumbling.

"We'll learn some interesting things, and then we'll come back to
reality. Honest!"

The class went back to their seats, and the Wiz wiped the sweat off his
forehead. "Phew!" he said. "At least you're taking the
subject seriously."

"Down with Wick rotation!" yelled Oz.

"Hey!" said the Wiz. "Who said that?" He spotted Oz
crouched behind a desk. "Aha, so there's the
troublemaker! There's the fellow who's been asking all those
questions in your voice, Toby!"

Toby turned and saw Oz quivering with fear. "Why, you little...."

Oz got up and made a break for the door. Furious, Toby chased him out
of class and down the hallway.

"I'll deal with them later," said the Wiz. "For now, let
me finish today's lecture. If we formally replace t by -it, the
standard Lorentzian metric on the cylinder becomes Riemannian, and
Schroedinger's equation

d psi / dt = - iH psi

becomes the heat equation

d psi / dt = - H psi.

Instead of time evolution being described by the one-parameter unitary
group exp(-itH), it's described by the one-parameter semigroup exp(-tH).
By "semigroup", I mean that exp(-tH) is only a nice bounded operator
for nonnegative times t. Basically, what this strange trick amounts to
is studying thermal physics rather than quantum physics... but they are
related by many useful analogies.

Generalizing this idea to the case of an arbitrary Riemannian metric on
the cylinder, we can define a theory where the Hilbert space for a
circle is L2(R) and the time evolution operator from
one slice S to another slice S':

much as in a TQFT. So one question is: can we extend this idea to more
general cobordisms, allowing topology change? Next week, we'll see that
the answer is yes. We'll get something a lot like a TQFT, but where
any cobordism M must be equipped with an area before we can define
Z(M). This sort of theory isn't completely background-free: the area
plays the role of a background structure. But as we've seen, it's much
closer to being background-free than your average quantum field theory
formulated with the help of a background metric. And later we'll see
how to modify this idea to get some truly background-free theories...
which don't involve this "Wick rotation" idea that you
hate so much."

Then the Wizard walked out in search of Oz and Toby, who could be
heard scuffling somewhere in the courtyard below.